The analogue of Bott's Decomposition Lemma yields
\(E^{p,q}_1 = H_\delta ^p \left(\bar{\Omega }^q (G^p)_n\right) \cong H_{\square }^{p-q} \left(G; S^q(\mathfrak {g}^*)_n\right) ,\) 

where \(H_\square ^*\)  refers to the cohomology of group germs,
which by its very definition (see [1]}) in the case at hand is the cohomology of
the complex \(\lbrace \bar{\Omega }^0 \left(G^p ; S^q( \mathfrak {g}^*)_n\right), \delta \rbrace \) .
Furthermore, the counterpart of van Est's isomorphism
for this case was proved by Świerczkowski [1]} and it gives
the isomomorphism
\( H_{\square }^{i} \left(G; S^q( \mathfrak {g}^*)_n\right) \cong H^{i} \left(\mathfrak {g}; S^q( \mathfrak {g}^*)_n\right).\) 

Since \(\, H^{\ast } \left(\mathfrak {g}; S^q(\mathfrak {g}^*)_n\right) \cong H^{\ast } (\mathfrak {g}; {\mathbb {R}}) \otimes S^q( \mathfrak {g}^*)_n^{G}\) , it follows that
\(H^{\ast } (\mathfrak {g}; {\mathbb {R}}) \otimes S( \mathfrak {g}^*)_n^{G}\cong E [h_1, h_2, \ldots , h_n] \otimes P_n [c_1, c_2, \ldots , c_n]\) 

After identifying the first differential as being
\(d_1 u_i=c_i\)  for \(1\le i \le n\) , i.e. showing that \(E_1\cong W_n\) ,
one can continues as in the first proof in §REF .
Again, the more satisfying argument follows the pattern of the second proof in §REF ,
with the maps
\(\varphi _m\)  and \(\psi _m\)  replaced by  
\( \varphi _m^{\rm germ}: W_1(\mathfrak {g})/{\mathcal {F}}_1^{m+1} \rightarrow \bar{\Omega }^*(\bar{N}G)/{\mathcal {F}}^{m+1}\) 
and
\( \psi _m^{\rm germ}: \bar{\Omega }^*(NG)_m \rightarrow \bar{\Omega }^*(\bar{N}G)/{\mathcal {F}}^{m+1}\) .
The proof that these maps induce isomorphisms of spectral sequences goes
along the same lines as in
[3]}, with the necessary modifications similar to those used by Świerczkowski
to adapt van Est's proof for the bicomplex in [4]} to the bicomplex
in [1]}.
